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Average properties of random walks on Galton-Watson trees. (English) Zbl 0894.60065
Summary: We study the \(\lambda\)-biased random walk on Galton-Watson trees by the Dirichlet principle and a formula of mean exit time of a Markov chain. We prove that the average of escaping probability and mean exit time are bounded by the counterparts of the corresponding random walks on \(\{0,1,2,\dots\}\). In particular we partially verify the recent conjecture of R. Lyons, R. Pemantle and Y. Peres on the upper bound of the speed of \(\lambda\)-biased random walk on Galton-Watson trees [Probab. Theory Relat. Fields 106, No. 2, 249-264 (1996; Zbl 0859.60076)].

60G50 Sums of independent random variables; random walks
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
05C80 Random graphs (graph-theoretic aspects)
Zbl 0859.60076
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